A sequel to the first six books of the Elements of Euclid, containing an easy introduction to modern geometry, with numerous examples. By John Casey.

BOO VI.

159

98. If a line of constant length slide between two fixed lines,
the locus of the centre of instantaneous rotation is a circle.
99. If two sides of a A given in species and magnitude slide
along two fixed Os, the envelope of the third side is a circle.
(BOBILLIER).
100. If the lengths of the sides of the A in Ex. 99 be denoted
by a, b, c, and the radii of the three Os by a, B, y; then
aa ~+ bB ~ cy = twice the area of the A, the sign + or - being
used according as the Os touch the sides of the A internally or
externally.
101. If five quadrilaterals be formed from five lines by omitting
each in succession, the lines of collinearity of the middle points
of their diagonals are concurrent. (H. Fox TALBOT.)
102. If D, D' be the diagonals of a quadrilateral whose four
sides are a, b, c, d, and two of whose opposite angles are 0, 0', then
D2 D'2 = a2 c2 + b2d 2 - 2abod cos (0 + 0').
103. If the sides of a A ABC, inscribed in a (, be cut by a
transversal in the points a, b, o. If a, B, y denote the lengths of
the tangents from a, b, c to the 0, then af3y = Ab. Be. Ca.
104. If a, b, c denote the three sides of a A, and if a, B, y denote the bisectors of its angles,
8 ab. s. area
'y = (a+ b) (b + c) (c a)
105. If a A ABC circumscribed to a ( be also circumscribed
to another A A'B'C', and in perspective with it, the tangents from
the vertices of A'B'C' will meet its opposite sides in three collinear
points.
106. If two sides of a triangle be given in position, and its
area given in magnitude, two points can be found at each of
which the base subtends an angle of constant magnitude.
107. If two sides of a triangle and its inscribed circle be given
in position, the envelope of its circumscribed circle is a circle.MANNHEIM.
108. If the circumference of a circle be divided into an uneven
number of equal parts, and the points of division denoted by the
indices 0, 1, 2, 3, &c., then if the point of the circle diametrically opposite to that whose index is zero be joined with all the
points in one of its semicircles, the rectangle contained by the
chords terminating in the points 1, 2, 4, 8... is equal to the
power of the radius denoted by the number of chords.